# An explicit construction of the universal division ring of fractions of $E⟨⟨x_{1},…,x_{d}⟩⟩$

### Andrei Jaikin-Zapirain

Universidad Autónoma de Madrid, Spain

## Abstract

We give a sufficient and necessary condition for a regular Sylvester matrix rank function on a ring $R$ to be equal to its inner rank $ρ_{R}$. We apply it in two different contexts.

In our first application, we reprove a recent result of T. Mai, R. Speicher and S. Yin: if $X_{1},…,X_{d}$ are operators in a finite von Neumann algebra $M$ with a faithful normal trace $τ$, then they generate the free division ring on $X_{1},…,X_{d}$ in the algebra of unbounded operators affiliated to $M$ if and only if the space of tuples $(T_{1},…,T_{d})$ of finite rank operators on $L_{2}(M,τ)$ satisfying [ \sum_{i=1}^d [T_k,X_k]=0, ] is trivial.

In our second and main application we construct explicitly the universal division ring of fractions of $E⟨⟨x_{1},…,x_{n}⟩⟩$, where $E$ is a division ring, and we use it in order to show the following instance of pro-$p$ Lück approximation.

Let $F$ be a finitely generated free pro $p$-group, $F=F_{1}>F_{2}>⋯$ a chain of normal open subgroups of $F$ with trivial intersection and $A$ a matrix over $F_{p}[[F]]$. Denote by $A_{i}$ the matrix over $F_{p}[F/F_{i}]$ obtained from the matrix $A$ by applying the natural homomorphism $F_{p}[[F]]→F_{p}[F/F_{i}]$. Then there exists the limit [ \displaystyle \lim_{i\to \infty} \frac{\mathrm {rk}_{\mathbb F_p} (A_i)}{|F

|} ] and it is equal to the inner rank $ρ_{F_{p}[[F]]}(A)$ of the matrix $A$.## Cite this article

Andrei Jaikin-Zapirain, An explicit construction of the universal division ring of fractions of $E⟨⟨x_{1},…,x_{d}⟩⟩$. J. Comb. Algebra 4 (2020), no. 4, pp. 369–395

DOI 10.4171/JCA/47